Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3508 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(13,14,18,23)(15,24,26,28)(16,22,17,21)(19,27,20,25);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,25)(18,26)(20,23)(21,28)(22,24);(6,9)(10,12)(16,17)(21,22)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,27)(17,25)(18,26)(19,23)(21,24)(22,28)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72279881, #72280101, #72280250, … |
| maximal normal subgroup of | #72279881, #72280101, #72280250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311177);